The sp_dropserver / sp_addserver method works only for the DEFAULT instance though. On the default instance only, by using sp_dropserver / sp_addserver you can change the server name reported by @@SERVERNAME. You would also have to change your machnine name if you wanted remote clients to be able to connect to that instance using that name.
For a NAMED (non-default) instance it isn't so easy. The supported method is to uninstall / re-install. There are registry hacks to rename a default instance, but you may prefer the safer option of re-installing: